Chromatic Bases for Symmetric Functions
نویسندگان
چکیده
In this note we obtain numerous new bases for the algebra of symmetric functions whose generators are chromatic symmetric functions. More precisely, if {Gk}k>1 is a set of connected graphs such that Gk has k vertices for each k, then the set of all chromatic symmetric functions {XGk}k>1 generates the algebra of symmetric functions. We also obtain explicit expressions for the generators arising from complete graphs, star graphs, path graphs and cycle graphs.
منابع مشابه
Chromatic classical symmetric functions
In this note we classify when a skew Schur function is a positive linear combination of power sum symmetric functions. We then use this to determine precisely when any scalar multiple of a skew Schur function is the chromatic symmetric function of some graph. As a consequence we prove that of the classical bases for symmetric functions only certain scalar multiples of the elementary symmetric f...
متن کاملLollipop and Lariat Symmetric Functions
We compute an explicit e-positive formula for the chromatic symmetric function of a 3 lollipop graph, Lm,n. From here we deduce that there exist countably infinite distinct e-positive, and 4 hence Schur-positive, bases of the algebra of symmetric functions whose generators are chromatic 5 symmetric functions. Finally, we resolve 6 conjectures on the chromatic symmetric function of a 6 lariat gr...
متن کاملA note on non-broken-circuit sets and the chromatic polynomial
1. Definitions and a main result 2 1.1. Graphs and colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3. Chromatic symmetric functions . . . . . . . . . . . . . . . . . . . . . 5 1.4. Connected components . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5. Circuits and broken circuits . . . . ...
متن کاملMacMahon Symmetric Functions, the Partition Lattice, and Young Subgroups
A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. In this article, we show that the MacMahon symmetric functions are the generating functions for the orbits of sets of functions indexed by partitions under the diagonal action of a Young subgroup of a symmetric group. We define a MacMahon chr...
متن کاملPower Sum Expansion of Chromatic Quasisymmetric Functions
The chromatic quasisymmetric function of a graph was introduced by Shareshian and Wachs as a refinement of Stanley’s chromatic symmetric function. An explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing the chromatic quasisymmetric function of the incomparability graph of a natural unit interval order in terms of power sum symmetric functions, is proven. The proof use...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Electr. J. Comb.
دوره 23 شماره
صفحات -
تاریخ انتشار 2016